Manifolds and Moduli Spaces

部分研究在这个补助金下进行将关注流形和他们的模的研究。 流形是空间的一个广义概念，在数学和科学中随处可见。 流形的定义性质是它可以由有限数量的参数局部参数化（例如，地球表面可以由两个参数局部参数化，即经度和纬度）。 一个经典的主题代数拓扑，加拉修斯的研究领域，是分类的流形：如何决定是否两个流形是同构的，以及如何写一个列表的所有可能的流形？ 在这里进行的研究涉及的相关问题的分类流形可能会有所不同的家庭：什么是他们的对称性和什么是他们的变形。 在这项资助下进行的研究将采用新的方法来研究这些经典和重要的问题。 在该资助下进行的另一部分研究涉及使用同伦理论的方法研究数论中产生的物体变形。拟议的活动将继续发展一个相对较新的数学领域，其中同伦理论用于研究各种模空间。 一个重要的早期结果，这方面的解决方案，由马德森和韦斯，一个猜想的芒福德。 从那时起，已经发生了重大的新进展，其中许多，如1995年Hatcher猜想的解决方案，都是由于首席研究员。 这门学科是代数拓扑学和其他数学分支的交叉，在代数几何、辛几何和可能的理论物理中有预期的应用。 该项目的另一部分涉及同伦理论思想在数论中的应用。 在导出的环境中工作是同伦理论中一个非常熟悉的想法，并且经常被成功地引入代数和其他领域。 最近，使用派生设置的变形的研究受到了很大的关注。 该项目将开发和应用这些技术的伽罗瓦表示，在数论中具有根本重要性的对象。

Parts of the research conducted under this grant will concern the study of manifolds and their moduli. Manifolds are a generalized concepts of space, appearing all over mathematics and science. The defining property of a manifold is that it is locally parametrized by a finite number of parameters (for example, the surface of the earth can locally be parametrized by two parameters, namely longitude and latitude). A classical topic in algebraic topology, Galatius' field of research, is the classification of manifolds: How does one decide whether two manifolds are isomorphic, and how does one write a list of all possible manifolds? The research conducted here concerns the related question of classifying how manifolds may vary in families: what are their symmetries and what are their deformations. The research conducted under this grant will apply new methods to study these classical and important questions. Another part of the research conducted under this grant concerns deformations of objects arising from number theory, studied using methods from homotopy theory.The proposed activity will continue the development of a relatively new area of mathematics in which homotopy theory is used to study various moduli spaces. An important early result this area is the solution, by Madsen and Weiss, of a conjecture of Mumford. Significant new developments have happened since then, many of which, such as the solution to a 1995 conjecture of Hatcher, are due to the principal investigator. This subject lies in the overlap between algebraic topology and other branches of mathematics, with expected applications in algebraic geometry, symplectic geometry, and possibly theoretical physics. Another part of the project concerns applications of homotopy theoretic ideas in number theory. Working in a derived setting is a very familiar idea in homotopy theory, and has often been successfully imported into algebra and other fields. Recently, the use of a derived setting for the study of deformations has received much attention. The project will develop and apply these techniques to Galois representations, objects of fundamental importance in number theory.